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A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 53 (8 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
Petri Nets And Step Transition Systems
 International Journal of Foundations of Computer Science
, 1992
"... Labelled transition systems are a simple yet powerful formalism for describing the operational behaviour of computing systems. They can be extended to model concurrency faithfully by permitting transitions between states to be labelled by a collection of actions, denoting a concurrent step. Petri ne ..."
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Cited by 43 (1 self)
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Labelled transition systems are a simple yet powerful formalism for describing the operational behaviour of computing systems. They can be extended to model concurrency faithfully by permitting transitions between states to be labelled by a collection of actions, denoting a concurrent step. Petri nets (or Place/Transition nets) give rise to such step transition systems in a natural way  the marking diagram of a Petri net is the canonical transition system associated with it. In this paper, we characterize the class of PNtransition systems, which are precisely those step transition systems generated by Petri nets. We express the correspondence between PNtransition systems and Petri nets in terms of an adjunction between a category of PNtransition systems and a category of Petri nets in which the associated morphisms are behaviourpreserving in a strong and natural sense.
Applications of Linear Logic to Computation: An Overview
, 1993
"... This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, li ..."
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Cited by 41 (3 self)
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This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...
ZeroSafe Nets: Comparing the Collective and Individual Token Approaches
"... The main feature of zerosafe nets is a primitive notion of transition synchronization. To this aim, besides ordinary places, called stable places, zerosafe nets are equipped with zero places, which in an observable marking cannot contain any token. This yields the notion of transaction: a basic ..."
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Cited by 40 (20 self)
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The main feature of zerosafe nets is a primitive notion of transition synchronization. To this aim, besides ordinary places, called stable places, zerosafe nets are equipped with zero places, which in an observable marking cannot contain any token. This yields the notion of transaction: a basic atomic computation, which may use zero tokens as triggers, but defines an evolution between observable markings only. The abstract counterpart of a generic zerosafe net B consists of an ordinary P/T net whose places are the stable places of B, and whose transitions represent the transactions of B. The two nets offer both the refined and the abstract model of the same system, where the former can be much smaller than the latter, because of the transition synchronization mechanism. Depending on the chosen approach  collective vs individual token philosophy  two notions of transaction may be defined, each leading to different operational and abstract models. Their comparison is fully dis...
The Stone gamut: A coordinatization of mathematics
 In Logic in Computer Science
, 1995
"... We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional space we call the Stone gamut. The Stone ..."
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Cited by 30 (13 self)
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We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional space we call the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from −1 for sets to 1 for complete atomic Boolean algebras (CABA’s), and vertically by complexity of language. Complexity 0 contains only sets, CABA’s, and the inconsistent empty set. Complexity 1 admits noninteracting setCABA pairs. The entire Stone duality menagerie of partial distributive lattices enters at complexity 2. Groups, rings, fields, graphs, and categories have all entered by level 16, and every category of relational structures and their homomorphisms eventually appears. The key is the identification of continuous functions and homomorphisms, which puts StonePontrjagin duality on a uniform basis by merging algebra and topology into a simple common framework. 1 Mathematics from matrices We organize much of mathematics into a single category Chu of Chu spaces, or games as Lafont and Streicher have called them [LS91]. A Chu space is just a matrix that we shall denote =, but unlike the matrices of linear algebra, which serve as representations of linear transformations, Chu spaces serve as representations of the objects of mathematics, and their essence resides in how they transform. This organization permits a general twodimensional classification of mathematical objects that we call the Stone gamut 1, distributed horizontally by ∗This work was supported by ONR under grant number N0001492J1974. 1 “Spectrum, ” the obvious candidate for this appliction, already has a standard meaning in Stone duality, namely the representation of the dual space of a lattice by its prime ideals. “A
On the Semantics of Place/Transition Petri Nets
, 1992
"... Abstract. In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical “token game”, one can model the behaviour of Petri nets via nonsequential processes, via unfolding constructions, which provide formal relationships between nets an ..."
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Cited by 22 (10 self)
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Abstract. In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical “token game”, one can model the behaviour of Petri nets via nonsequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. More precisely, we introduce the new notion of decorated processes of Petri nets and we show that they induce on nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net N can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification.
Chu spaces and their interpretation as concurrent objects
, 2005
"... A Chu space is a binary relation =  from a set A to an antiset X defined as a set which transforms via converse functions. Chu spaces admit a great many interpretations by virtue of realizing all small concrete categories and most large ones arising in mathematical and computational practice. Of pa ..."
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Cited by 21 (0 self)
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A Chu space is a binary relation =  from a set A to an antiset X defined as a set which transforms via converse functions. Chu spaces admit a great many interpretations by virtue of realizing all small concrete categories and most large ones arising in mathematical and computational practice. Of particular interest for computer science is their interpretation as computational processes, which takes A to be a schedule of events distributed in time, X to be an automaton of states forming an information system in the sense of Scott, and the pairs (a, x) in the =  relation to be the individual transcriptions of the making of history. The traditional homogeneous binary relations of transition on X and precedence on A are recovered as respectively the right and left residuals of the heterogeneous binary relation =  with itself. The natural algebra of Chu spaces is that of linear logic, made a process algebra by the process interpretation.
ZeroSafe Nets, or Transition Synchronization Made Simple
 PROC. OF EXPRESS’97
, 1997
"... In addition to ordinary places, called stable, zerosafe nets are equipped with zero places, which in a stable marking cannot contain any token. An evolution between two stable markings, instead, can be a complex computation called stable transaction, which may use zero places, but which is atomic w ..."
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Cited by 17 (13 self)
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In addition to ordinary places, called stable, zerosafe nets are equipped with zero places, which in a stable marking cannot contain any token. An evolution between two stable markings, instead, can be a complex computation called stable transaction, which may use zero places, but which is atomic when seen from stable places: no stable token generated in a transaction can be reused in the same transaction. Every zerosafe net has an ordinary PlaceTransition net as its abstract counterpart, where only stable places are maintained, and where every transaction becomes a transition. The two nets allow us to look at the same system from both an abstract and a refined viewpoint. To achieve this result no new interaction mechanism is used, besides the ordinary tokenpushing rules of nets. The refined zerosafe nets can be much smaller than their corresponding abstract P/T nets, since they take advantage of a transition synchronization mechanism. For instance, when transactions of unlimited l...
Between Functions and Relations in Calculating Programs
, 1992
"... This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made ..."
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Cited by 15 (4 self)
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This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made using Ruby. The first is that the notion of a program being an implementation of a specification has never been made precise. The second is to do with types. Fundamental to the use of type information in deriving programs is the idea of having types as special kinds of programs. In Ruby, types are partial equivalence relations (pers). Unfortunately, manipulating some formulae involving types has proved difficult within Ruby. In particular, the preconditions of the `induction' laws that are much used within program derivation often work out to be assertions about types; such assertions have typically been verified either by informal arguments or by using predicate calculus, rather than by ap...